Continuous casting plant

ABSTRACT

A continuous casting plant comprising guiding, bending and/or straightening rollers, which are arranged along a curve corresponding to the differential equation   wherein phi &#39;&#39; (xj) is a function of the elongation alteration, or change in elongation RE is equal to the circular arc radius adjacent to the bending and/or straightening zone, respectively, and XE is the vertical projection of the bending zone and the horizontal projection of the straightening zone, respectively, and wherein xj is a position coordinate in a Cartesian coordinate system, whose origin lies, in each case, at the beginning of the transition curve and whose x-axis is a tangent to the transition curve, on the outer side of the strand during bending and on the inner side of the strand during straightening.

O Unlted States Patent 11 1 [111 3,893,503

Eibl 1 1 July 8, 1975 CONTINUOUS CASTING PLANT [57] ABSTRACT lnvemol'l Erich Eibl, UHZ, Austria A continuous casting plant comprising guiding, bend- [73] Assignee vereinige osterreichische Eisem ing and/or straightening rollers, which are arranged und Smhlwerke Alpine Monm along a curve corresponding to the differential equa- Aktiengesellschaft, Linz, Austria [22] Filed: Oct. 18, 1973 21 Appl. No.: 407,473 y" (9,).

of E 1 [30] Foreign Application Priority Data July 24, 1973 Austria 6511/73 herein o (x is a fun ti n of the elongation alteration, or change in elongation R is equal to the circu- [52] US. Cl 164/282; 164/82 lar arc radius adjacent to the bending and/or straight [51] Int. Cl B22d 11/12 ening zone, respectively, and X is the vertical projec- [58] Field of Search 164/82, 282 tion of the bending zone and the horizontal projection of the straightening zone, respectively, and wherein x, [56] References Cited is a position coordinate in a Cartesian coordinate sys- UNITED STATES PATENTS tem, whose origin lies, in each case, at the beginning 3 290 74' 12/1966 olsson [64/282 x of the transition curve and whose x-axis is a tangent to 332493l 6/1967 54/82 the transition curve, on the outer side of the strand 3:645:323 2 1972 Vogt et a1. 164/282 during bending on the inner Side Of the Strand 3,729,048 4 1973 Gelfenbein etal. 164/282 during stralghtemng- 3,752,210 8/1973 Gallucci et al 164/82 Primary Examiner-R. Spencer Annear Attorney, Agent, or Firm-Brumbaugh, Graves, Donohue & Raymond 21 Claims, 20 Drawing Figures EMFWFMUL 8 ms SHEET FIG? SHEU

a ITFUJUL 8 I975 PRIOR ART F'IG I3 PRIOR ART SHEET PRIOR ART F1615 PRIOR ART 954 ,0

SHEET FIG. 20

m Ilffl I CONTINUOUS CASTING PLANT The invention relates to a continuous casting plant with a secondary cooling zone arranged to follow the mold and provided with rollers for guiding, bending and/or straightening the strand, wherein the bending rollers are arranged along a transition curve from the vertical to the circular arc and the straightening rollers are arranged along a transition curve from the circular arc to the horizontal.

Such continuous casting plants in which the strand is bent into the horizontal have a lower overall height as compared with vertical plants. If one wants to use a straight mold instead of a curved mold, which has metallurgical advantages, between the mold or a short vertical guiding part following the mold, respectively, and the guide path having the form of a circular arc of e.g. a diameter of 8 to 10 m, a curved intermediate piece following a transition curve must be provided, so as to avoid sudden elongation stresses at the outer side of the strand, whose core is still liquid. Therefore it has been proposed from various parts, see eg US. Pat. No. 3,290,741, Austrian Pat. No. 231,629 and Austrian Pat. No. 244,522 to form the transition curve in such a way that the radius of curvature is gradually or stepwisely increasing, e.g. resembling a hyperbola, a parabola, an ellipse or a clothoid.

Also the straightening zone at the transition from the circular arc-shaped part of the secondary cooling zone into the horizontal part of the conveying path should be formed along a transition curve, so as to avoid excessive elongations at the inner side of the strand.

In spite of the known propositions in this field it has hitherto not been achieved to avoid the undesired discontinuity points, because in all curves of second order, such as a circle, ellipse, hyperbola, parabola, a sudden (jump-like) change of the radius of curvature necessarily occurs at the transition from a tangent (radius of curvature infinite) onto a point of the curve (radius of curvature finite). Moreover, the known propositions for the formation of the transition arc have not taken into consideration the respective elongation alteration A D, i.e. the derivative of the elongation. So far, the necessity of adhering to a maximumly permissible elongation alteration has not been recognized.

It is the object of the invention to form the transition curve in the bending and/or straightening zone, in which the bending rollers and straightening rollers, respectively, are arranged, so as not to exceed at any time a certain permissible elongation alteration at the outer side of the strand or at a certain spot between the outer side of the strand and the neutral fiber, i.e. along a selectable line within this area, during bending, and so as not to exceed a certain permissible elongation alteration at the inner side of the strand or at a certain spot between the inner side of the strand and the neutral fiber, during straightening. Moreover, the final radius at the end of the bending zone and the initial radius at the beginning of the straightening zone are to be absolutely equal to the radius of curvature of the guiding part, which has the form of a circular arc and being arranged between the bending zone and the straightening zone.

According to the invention a this object is achieved in a plant of the above described type in that the bending and straightening rollers, respectively, are arranged along a curve which corresponds to the differential equation:

v'"= 'u d) i 5 f (m m- ,q5(x being a function of the elongation alteration, (change in elongation) which, along the extents of the bending and straightening zones, respectively, shows at first an increasipg course, then reaches the maximum of the permissible elongation alteration and then again has a declining course, R being equal to the radius of the circular arc at the end of the bending zone and the beginning of the straightening zone, respectively, and X being the vertical projection of the bending zone or the horizontal projection of the straightening zone, respectively, and x, being a position coordinate in a Cartesian coordinate system, whose origin lies at the beginning of the transition curve and whose x-axis forms a tangent to the transition curve to the outer side of the strand during bending, and to the inner side of the strand during straightening. The invention therefore is based on the finding that what matters is the elongation alteration d '(x which is a derivation from (x,-), a function proportionate to the elongation. As in steel in the transition phase liquid-solid, the deformation resistance depends on the deformation rate, only alterations of the elongation cause tensions and only this alteration is important for the danger of crack formation. If the derivation of the elongation d '(x,-) exceeds a limit value, cracks occur at the point concerned. The course of d '(x,-) must therefore occur in such a way that this limit value is not exceeded during bending and straightening, respectively.

Advantageously the maximum elongation D is assumed with 50d/R d being the thickness of the stand and R; as defined above.

According to the invention, the rollers for bending and straightening the strand are arranged at points which result from a course, free from marked jumps, of the function of the elongation alteration or change in elongation (x The course is preferably polygonal, in particular trapezoidal, but may also be in the form of a circular arc, a parabola or a similar shape.

The invention is described in more detail by way of examples with reference to the accompanying drawings.

FIGS. 1, 2, 3 and 4 relates to a first embodiment;

FIG. I is a schematic side view, not to scale, of a bending device of the present invention.

FIG. 2 illustrates the bendingand supporting forces acting on a cast strand.

FIG. 3 is a graph corresponding or proportionate to a standardized (i.e. the maximum of the function equals 1 course of the momentum line, or to the elongation alteration at the outer side of the strand, or to the third derivation y' (x each of them referring to the x-axis of a coordinate system, whose origin lies at the beginning of the bending device and at the outer side of the strand, and y (x being the function for the geometrical course of the bent strand skin.

FIG. 4 is a graph, which shows the effective course of the elongation and of the elongation alteration of the outer strand skin, each being referred to the above mentioned coordinate system.

FIGS. 5, 6, 7 and 8 relate to a second embodiment of the invention, wherein the FIGS. 5, 6 are illustrations similar to FIGS. 1 and 2, and the FIGS. 7 and 8 correspond to the FIGS. 3 and 4.

FIGS. 9, l and 11 are illustrations similar to FIGS. 3 and 7, but for different cases of stress to which the strand is exposed during bending in the bending device of the present invention.

FIGS. l2, I3, 14 and 15 are illustrations similar to FIGS. 4 and 8 and refer to prior art, FIG. 12 corresponding to the geometrical course of the bent strand skin following a circle, FIG. 13 following an ellipse, FIG. 14 following a common parabola, and FIG. 15 following a clothoid.

FIGS. 16, 17, 18 and 19 refer to a third embodiment of the invention, i.e. to a straightening device, FIGS. 16, 17 being similar to FIGS. and 6, and FIGS. 18 and 19 being similar to FIGS. 7 and 8.

FIG. 20 is a side view of a strand that is first bent in a bending device of the present invention and then straightened in a straightening device of the present invention and serves to illustrate the position of the coor dinate system in the bending device and the straightening device, respectively.

In FIGS. 1 and 2, the straight strand, which is being introduced into the bending device of the present invention, is denoted with I. It is a continuously cast strand, which is drawn from a water-cooled mold and has a rectangular cross section and a thickness of d 200 mm, and which, after leaving the mold has a surface temperature of approx. l400C and a strand skin thickness of approx. 30 mm. Below the mold (not shown), a vertical straight roller guiding with rollers arranged in pairs and lying opposite each other is provided for supporting and guiding the strand 1, the last pair of rollers of this straight strand guiding being marked in dotted lines and being denoted with 2, 3.

The bending device of the present invention comprises nine non-driven roller pairs, the rollers 4, 7, 9, l3, l7, I9, 20 pertain to the device causing the bending of the strand 1, while the rollers 5, 6, 8, 10, 11, 12, 14, l5, l6, I8, 21 merely have a supporting function, i.e. they shall counteract a bulging of the strand as a consequence of the ferrostatic pressure. All the rollers 4 to 21 are not driven and are freely rotatable. The strand I leaves the bending device with a radius R 8000 mm and runs in the direction of the arrow into the strand guiding, which has the form ofa circular arc and consists also of roller pairs, the first of which being illustrated in dotted lines and being denoted with 22 and 23.

The bending rollers pertaining to the bending device proper are the rollers 7, 9, 13, 17, 19, whereas the rollers 4 and 20 are the supporting rollers that accommodate the reaction forces. The centers and the axes of these rollers are denoted with 7", 9", 13", 17", 19", and the tangent points or the generatrices', respectively, of the surface area of these rollers which touch the strand skin are denoted with 7', 9', 13', 17', 19'. The remaining rollers, which only accommodate the ferrostatic pressure, touch the strand skin in the points or lines 5', 6,8',10', 11', 12', 14,15, 16,18', 21' and their centers or axes are analogously denoted with 5", 6!! 11,1 1! 1111,1211, 14H, H, 16H 18!! 21!!- All the rollers 4 to 21 just like the rollers 2, 3 and 22, 23, respectively are arranged equidistantly and the tangent points 4', 5' to 21' lie on curve paths 24, 25, which correspond to the geometrical course of the outer and inner strand skin, respectively. The curve 24 corresponds to a function y(x whose coordinates are to be defined, the origin of the coordinate system x, y lying in the point 4'. The y-axis is denoted with 26 and marks the beginning of the transition zone. The xaxis is denoted with 27 and is a tangent to the curve 24 in the point 4'. The direction of the positive x-axis runs in the direction of the movement of the strand, while the direction of the positive y-axis 26 runs perpendicularly towards the strand interior. Therefore, 26 simultaneously denotes the plane through the roller axes 4", 5" and through the tangent points 4', 5', and the perpendicular to the tangent in the tangent points 4', 5 of the curves 24, 25. Analogously, the corresponding planes and perpendiculars, respectively, through the subsequent pairs of rollers are denoted with 28, 29, 30, 3 l, 32, 33, 34, 35. In the tangent point 4', the curve 24 has a radius of curvature R R and the radii of curvature in the subsequent tangent points 6', 8' to 20' are analogously denoted with R R,, R,,,, R R R R and R wherein R R 8000 mm. The angles of inclination of the planes or the perpendiculars 28 to 35 through the tangent points 6', 8' to 20' to the y-axis 26 are denoted with 11 a a 0: a 01 a and 0 and must also be defined for the bending device of the present invention.

For construction, the following data are given: the x-coordinates of the tangent points 6', 8, 12', 16', 18', 20', or the distances of these points from each other. The distance A is to measure 200 mm and thus the overall length of the curve 24, when projected into the x-axis 27 is X 1600 mm. As shown magnified in FIGS. 1 and 2, the distances between the tangent points 4', 6' and 8', and 16', 18' and 20', respectively, are equal, i.e. each being A 200 mm, and the distance between the tangent points 8', 12' and 16' is twice as big, i.e. each being 2A 400 mm.

FIG. 2 is a schematical illustration of the vectors of the bending forces P P P,;,, P P acting upon the inner side of the strand and of the corresponding supporting forces (reaction forces) P P The lines of influence of these vectors lie in the planes 28, 29, 31, 33, 34, or perpendiculars 26, 35, respectively. FIG. 2 also illustrates a tangent 27 for the curve 25, displaced parallel in relation to the x-axis 27 and running through the tangent point 5'.

In FIG. 3, the x-coordinate of the points 4', 6, 8', 12', 16', 18', 20' are plotted on the x-axis. They are denoted as x,, x x x x,,, X x On the y-axis, the measurements 1.000, 0.667 and 0.375 are plotted on an arbitrarily chosen graduation, which figures constitute prorated values and in this specific case ycoordinates for the points x x and x and x and x,,,' respectively, for forming a mirror-symmetrical curve 42 with the points 4', 39, 37, 36, 38, 40, 41, the axis of symmetry running through the points 36 and Jr The curve 42 is drawn in broken lines and the area 43 enclosed between the curve and the x-axis is hatched.

The course of the curve 42 has the following three meanings:

In the embodiment shown, the curve 42 takes a polygonal course, similar to the course of a moment line, which would result from the statically determined load case illustrated in FIG. 2. Moreover, the course of the curve 42 is similar to the course of a curve that corresponds to the elongation alteration of a line element of the outer strand skin expressed in %/mm, measured along the x-axis 27. Third, the course of the curve 42 is similar to the course of a curve corresponding to the curvature alteration of the curve 24 or to the third de' rivative y' (.r,) of the function y(x respectively.

The following characteristics of the curve 42 are essential for the invention:

It has a first real zero point in the origin of the coordinate system 4', respectively 1: It first increases relatively strongly, then it becomes gradually flatter up to a maximum value 36. The second part of the curve from the maximum value 36 to the second real zero point in the point 41 is in this embodiment mirrorsymmetrical. After having reached its maximum value, the curve first descends relatively slightly, then it descends gradually more strongly to this second real zero point. The existence of two real zero points 4' and 41, respectively, at the beginning and at thetend of the curve 42, respectively, corresponding to the beginning and the end of the bending zone extending between the lines 26 and 35 related to the x-axis 27 with a maximum value 36 lying in between is the essence of the invention, from which one starts in order to find the geometrical locus of all the tangent points of the rollers, and the points of impact of the bending device, respectively, and to design the bending device of the present invention, the transition curve 24 from the straight line into the circular arc not following a curve known from prior art, resembling a circle, an ellipse, a parabola, a hyperbola or a clothoid, or a catenary. Furthermore, it is essential that the points 39, 37, 36, 38, 40 in this embodiment, where several individual forces P P P P P cause the bending are, though buckles, no discontinuities of the curve 42.

The area 43, expressed in mm, is an auxiliary value FLE, which is necessary for the further calculation. In order to obtain the course of the curve for the elongation alteration 2' (x,), which is denoted with 42' in FIG. 4 and illustrated true to scale, the curve 42 of FIG. 3, which is denoted as Q5 (x,), is lowered, i.e. it is transformed with the transforming factor k". The following equations apply:

R; FLE 3 k is a further transforming factor, which is later on used for the calculation of the curve 24, which follows the function y (.r,).

The ordinate values for the curve 42' and the points 39', 37, 36', 38, 40', respectively are obtained by multiplying the y-values of the curve 42 with the transforming factor k".

By integration of the curve 42' to dx, one obtains the curve 44 in FIG. 4, which corresponds to the elongation of a line element of the outer strand skin, expressed in and drawn true to scale. The %-graduation treme value 46 with 44", the one tangent coinciding with the x-axis 27 and the other tangent being denoted with 27m At the point where the curve 42 has its maximum value 36' a point of inflexion 45 is present in the curve 44.

in particular the following remarks referring to the calculation have to be made:

:{For the curvature of y (x,), the geometrical locus of the outer strand skin exposed to elongation in the bending device, the following applies:

R, being the respective radius of curvature. For a percentage of a line element of the outer strand skin in the bendingdevice or of the inner strand skin which is elongated in the straightening device the following applies:

d standing for the thickness of the strand, expressed in mm. In this embodiment, d 200 mm. From the equations 5 and 6, it results that the curvature K is proportionate to the elongation 209). Furthermore one can, without committing a considerable error, equate the curvature K with the second derivation y" (x which means that the following applies:

K yn i) Analogously, the alteration of the curvature K is proportionate to the elongation alteration 2' (x which means that the following applies:

and I t Kl v iylll 9 which means that the alteration of curvature is also proportionate to the third derivation of y (x These differential equations 1 to 9 are generally valid for the calculation of y (x,) and of the bending device, respectively.

In order to obtain the numerical value for the present embodiment, one has to start from equation 4. The following applies:

, thus obtaining a differential equation of third order for y(x,), which is easy to solve. Subsequently y" is obtained by integrating (equation 7) and by integrating once again In all the integration steps, one integrates from x 0, the origin of the coordinate system, as far as the current point x x, The final point x, is X E and in embodiment 1, x3 mm X 1n the present embodiment, all rollers have the same radius r 50 mm and the roller centers 4", 6", 8" to 20" of the rollers 4, 6, 8 to 20 on the outer strand skin, respectively, on the curve 24 are calculated from the coordinates x, and y,:

.r,= x, r. sin a,

y,= y; r. cos a,

The roller centers 7", 9" to 21" of the rollers 5, 7, 9 to 21 on the inner strand skin, respectively, of the curve 25 may be defined analogously from the coordinates x, and y,':

x, x, (d r) sina,

y,'=y,+(d+r) .cos a;

The coordinates of the curve 25, which corresponds to the geometrical locus of the inner strand skin, can be calculated from the following equations:

Y X d sina,

y y,+d.cosa,

The coordinates x,,,, y for the geometrical locus of the curvature centers of the curve 24 and 25 can be calculated from:

The results of the numerical calculation for the present embodiment are summarized in the following tables.

Table 1 contains the values for the elongation alteration and the total elongation on the outer strand skin, which correspond to the curves 42' and 44 in FIG. 4, and the radius of curvature R}.

In Table 2, the first two columns contain the coordinates for the curve 24, corresponding to the function y (x and the increase (slope) of this curve, expressed as y (x,), and tan 01,, respectively, is contained to the third column. The angles a a to 0: can be calculated immediately from the numerical values of column 3.

Table 3 contains in its upper part the coordinates of the rollers 4, 6, 8 to 20, according to the equations 13, 14 and in its lower part the coordinates of the opposite rollers on the inner side of the strand 1, according to the equations 15, 16. The term roller axis" is identical with roller center."

Table 1 Bending zone (FIG. 1-4) Distance on the elongation total radius R, x-axis alteration elongation mm mm 10 %/mm 0 (=x 0 0 w 50 0.124 0.00310 3 222 901 0.284 0.01241 865 810 150 0.372 0.02792 358 164 200 (=19 0.496 0.04963 201 489 250 0.593 0.07685 123 300 0.689 0.10888 91 836 350 0.785 0.14574 68 614 400 (=x 0.881 0.18740 53 360 450 0.936 0.23284 42 948 500 0.990 0.28099 35 588 550 1.045 0.33188 30 138 600 1.099 0.38547 25 942 650 1.153 0.44179 22 635 700 1.208 0.50081 19 968 750 1.262 0.56254 17 777 800 0 1.316 0.62696 15 950 850 1.2599 0.69135 14 464 900 1.2044 0.75296 13 281 950 1.1491 0.81179 12 318 1000 1.0938 0.86787 11 522 1050 1.038 0.92118 10 856 1100 0.984 0.97173 10 291 1150 0.928 1.0195 9 808 1200 (=Xm) 0.874 1.0646 9 393 1250 0.778 1.1059 9 043 1300 0.682 1.1424 8 754 1350 0.586 1.1740 8 516 1400 (=Xua) 0.490 1.2009 8 327 1450 0.368 1.2224 8 181 1500 0.245 1.2010 8 079 1550 0.123 1.2469 8 019 1600 (=X2n) 0 1.2500 8 000 Table 2:

Bending zone FIG. 1-4

Distance on Y, Y,

the x-axis mm (tana 0 (=x O O Table 2: -Continued Bending zone FIG. 1-4

Distance on Y, the x-axis mm (tana 50 0.000065 0.000005 100 0.001034 00000414 150 0.005236 0.0001396 200 (==x,,) 0.016546 0.000331 250 0.040379 0.000645 300 0.083525 0.001 1074 350 0.15399 0.001742 400 (=x,,) 0.26099 0.002573 450 0.41493 0003622 500 0.62713 0004906 550 0.90964 0.006437 600 1.275 0.008229 650 1.737 0.010296 700 2.310 0.012651 750 3.007 0.015309 800 (=x 3.845 0.018281 850 4.841 0.021578 900 6.009 0.025190 950 7.364 0.029103 1000 8.924 0.033039 1050 10.699 0.037777 1 100 12.706 0.042511 1 150 14.955 0.047490 1200 (=x 17.459 0.052702 1250 20.229 0.051296 1300 23.275 0.06375 1350 26.607 0.06954 1400 (=x 30232 0.07548 1450 34.157 0.08155 1500 38.388 0.08769 1550 42928 0.09391 1600 (=x,.,) =X 47.799 Y 0.10015 Table 3:

Bending zone (FIG. 1-4) position of the x, Y,

roller axes mm mm a) on the outer side roller 4 50 roller 6 200.02 49.98 roller 8 400.13 49.74 roller 10 600.41 48.72 roller 12 800.91 46.15 roller 14 1001.67 4l.05 roller 16 1202.63 32.47 roller 18 1403.76 l9.63 roller 20 1604.98 1.97

mm mm b) 0n the inner side roller 5 0 250 roller 7 199.92 250.02 roller 9 399.36 250.26 roller 1 1 597.94 251.27 roller 13 795.43 253.80 roller 991.68 258.78 roller 17 1186.84 267.11 roller 19 1381.18 297.52 roller 21 1575.09 296.54

In FIGS. 5 to 8, a modified embodiment of the continuous casting plant of the present invention is illustrated. The straight strand entering the bending zone is denoted with l and the rollers 2, 3, indicated in broken lines, are the last rollers of a straight strand guiding. The strand 1 again has a thickness of d 200 mm and is withdrawn from the bending device in the direction of the arrow and is further guided in a circular arc guiding, indicated by the roller pair 22, 23, the circular arc guiding having an outer radius R 8000 mm.

The bending device of the present invention comprises four rollers 47, 49, 50, 48, the rollers 49, 50 producing bending forces P P on the inner side of the strand and the roller 47, 48 causing reaction forces P P (FIG. 6). The bending rollers 49, 50 and the supporting rollers 47, 48 are not driven. Opposite the bending rollers 49, 50, guiding rollers 51, 52 are arranged, which do not take part in the proper bending of the strand, but counteract the ferrostatic pressure of the liquid strand core. The rollers 51, 52 may e.g. be omitted, if a completely solidified strand, e.g. a rail, is bent. The corresponding tangent points of the rollers are denoted with 47', 48', etc., analogous to the embodiment according to FIGS. 1 4, and similarly the roller centers or roller axes, respectively, are denoted with 47", 48", etc. The positive yaxis is denoted with 53, and 59 denotes the positive x-axis of a rectangular coordinate system for the curve 54 laid through the tangent point 47'. The curve 54 corresponds to the outer strand skin, and hence to the function y(x;), and the curve 55 corresponds to the equidistant inner strand skin. 56, 57, 58 are perpendiculars to the curve 54 in the tangent points 51, 52' and 48' and the pertaining radii of the curve 54 bear the denominations R R R The angles of inclination of the perpendicular in relation to the y-axis 53 are denoted with a The distance of the tangent point 51' projected onto the x-axis 59 is marked with B and measures 530 mm, the corresponding distance between the points 51 and 52' is C and measures mm and the distance between the points 52' and 48 is D and measures 570 mm. X therefore measures 1280 mm in this embodiment.

The calculation procedure and the equations to be used are the same as in the embodiment according to FIGS. 1 4.

The results are explained in more detail with reference to FIGS. 7 and 8 and to the numerical tables 4, 5 and 6.

According to the assumed load case with two bending forces P P the course of the curves 63 and 63', respectively, results to be in form of a trapezoid, i.e. two maximum values 60, 61 are present, which forms a zone, i.e. the zone of the maximum value between x and x which corresponds to the maximum value of the elongation alteration. At 47' and 62, there are again real zero points and the hatched area 64 enclosed between the curve 63 and the x-axis is FLE, i.e., as mentioned before, an auxiliary value for calculating the curve 63, which is drawn true to scale. The corresponding numerical values for the elongation alteration are contained in Table 4, and the numerical values for the total elongation may also be found there, by means of which the curve 65 in FIG. 8 was drawn. Between the curves 63' and 65, the same mathematical relationship exists as in the embodiment according to FIGS. 3 and 4. The curve 65 consists of a straight section 65', which coincides with a tangent in the x-axis 59 to the point 47', a section in form of a parabola that reaches as far as point 66, an inflectional tangent 65" between the points 66, 67, another section in form of a parabola from point 67 to the maximum value 68 and a straight section 65" parallel to the x-axis 59, which section coincides with the tangent 59' to the point 68.

Table 5 contains numerical values for the curve 54 and its ascent or slope, expressed as tangent of 01,-. The curve 54 represents the geometrical locus of the bent outer strand skin in the bending zone. At its beginning and at its end, the supporting forces P P act, while the bending forces P P act on the equidistant curve 55. The respective radius of curvature of the curve 54 can be seen from column 3 of table 4, and table 6 contains the coordinates x,", y," of the roller centers of 5 the rollers 47, 51, 52, 48 on the outer strand skin and the coordinates x,", y,"' of the two bending rollers 49,

50 on the inner strand skin.

Table 4:

Bending zone (FIG. 58)

distance on elongation total radius R;

the x-axis alteration elongation mm 10 mm (=x) 0 0 w 50 0.16 0.00406 2 460 679 100 0.33 0.01625 615 212 150 0.49 0.03657 273 456 200 0.65 0.06500 153 841 250 0.81 0.10155 98 476 300 0.97 0.14620 68 402 350 1.14 0.19894 50 267 400 1.30 0.25976 38 498 450 1.46 0.32864 30 428 500 1.62 0.40557 24 657 530 1.71 0.45558 21 950 550 1.71 0.48988 20 413 600 1.71 0.57558 17 373 650 1.71 0.66120 15 124 700 1.71 0.74675 13 391 710 (=x52) 1.71 0.76385 13 091 750 1.59 0.82983 12 051 800 1.44 0.90550 11 043 850 1.29 0.9736 10 278 900 1.14 1.034 9 669 950 0.99 1.087 9 197 1000 0.84 1.133 8.827 1050 0.69 1.171 8 540 1 100 0.54 1.202 8 322 1150 0.39 1.225 8 165 1200 0.24 1.240 8 062 1250 0.08 1.248 8 009 1280 (=x,,,) 0 1.250 8 000 Table Bending zone (FIG. 58)

Distance on y, y,

the x-axis mm (tana 0 (=x 0 0 50 0.000085 0.000007 100 0.00135 0.000054 150 0.00686 0.000183 200 0.02167 0.00433 250 005291 0.000846 300 010969 0.001462 350 020320 0.002322 400 034661 0.003465 450 0.55513 0.004933 500 0.84597 0.006765 5.30 (=x,,,) 1.06791 0.008056 550 1.238 0.009001 600 1.753 0.01166 650 2.412 0.01475 700 3.236 0.01828 710 (1152) 3.423 0.019032 750 4.247 002222 800 5.465 0.02656 850 6.909 003126 900 8.597 0.03628 950 10.542 0.04159 1000 12760 0.04714 1050 15.261 0.05291 1 100 18.054 0.05884 1150 21.147 0.06491 1200 24546 0.07108 1250 28256 0.07730 1280 (=x,,.)=X 30.632=Y 0.08105 Table 6:

Bending zone (FIG. 5 8) position of the x," Y," roller axes mm mm a) on the outer side roller 47 0 -50 roller 51 530.40 48.93 roller 52 710.95 -46 57 roller 48 1284.04 19.20

mm mm b) on the inner Side roller 49 527.99 251.06 roller 50 705.24 253.24

In FIGS. 9, 10, 11, further characteristic examples for the freely selectable course of the standardized moment line, according to different load cases of the strand in the bending zone of the present invention, are illustrated.

According to FIG. 9, four bending forces act upon the inner strand skin in the area of the points x x x x;,,. The corresponding supporting and reaction forces act upon the outer strand skin in the area of the points 1: x and a curve coarse drawn in broken lines results, characterized by a polygonal course 69' with a maximum value in the point 169, respectively at X if the magnitude of the bending forces is chosen in such a way that the resultant lies within this area. The inked curve 69 represents the ideal case as regards the bending stress acting upon the strand, i.e. when a constant load (uniform load) is present at the inner side of the strand, which arises when using continuous bending elements extending along the entire bending zone. In order to keep the friction between strand and supporting elements as low as possible, according to the invention, the maximum 169 is placed in the area of the last third of the entire bending zone. x therefore amounts to approx. of X The points 70, 71, 72 of the curve 69 are buckles, yet not discontinuities.

According to FIG. 10, only a single bending force acts at x and the supporting and reaction forces, respectively, act symmetrically thereto at X and x so that a triangular approximated curve 73' results having a maximum value 74. With an evenly distributed load for bending the strand, however, a symmetrical parabola 73 results, whose vertex is at the same time the maximum value and lies at 74.

According to FIG. 11, the maximum of a polygonal course 75 is characterized by a zone between the points 78, 76 and 79, which corresponds to the maximum value, while the points 77, 80 are buckles, yet no discontinuities. This is the ideal case of the design of the bending device of the present invention by using bending rollers for the transmission of the bending forces which act in the points x x x x x while the supporting forces and reaction forces, respectively, act at x and x If, however, a uniform bending load is assumed, a course in the form of a circular arc like the curve 75 results for the elongation alteration or change in elongation, the moment line and the curve corresponding to the third derivative y' (1,). The flat curve 75' and 75, respectively, following a great longitudinal extension (long bending zone) for the elongation alteration characterizes a, so to speak, mild stress on the strand, even if one has to put up with a greater friction because of the great length of the bending zone. On the other hand, according to FIG. 10, the

bending zone is relatively short and the friction is smaller, but the strand is bent in a very. short section, i.e. it is exposed to a relatively heavy stress and the stress is limited to a short zone. i

The following FIGS. 12 and I serve for a better understanding of the invention as compared with the prior art.

According to the prior art, e.g. to Austrian Pat. No. 231,629, it is known to use as a transition curve from a straight direction into a circular are, or vice versa, curves which correspond to circular arcs having stepwisely graduated radii of an ellipse, a parabola, a hyperbola, a clothoid or a catenary.

In an illustration analogous to FIGS. 4 and 8 but under the assumption that the transition curve (the geometrical locus of the outer strand skin) follows circular arcs of an ellipse, a parabola, or of a clothoid, FIGS. 12 to 15 result.

In FIG 12, the inked line 81 represents the curve of the elongation alteration 2' (x,-), which shows at the be ginning of the bending zone, at x a discontinuity from w to zero, characterized by the curve 81', and atthe end of the bending zone, at x a discontinuity from zero to characterized by the curve 81". In practice, this means that in an infinitely small zone, at x a sudden, abrupt, infinitely big elongation alteration occurs, while along the bending zone the elongation alteration is constantly zero and at the end of x, again an abrupt, infinitely big elongation alteration occurs in an infinitely small zone at the transition to the subsequent circular arc. The curve 82, shown in broken lines, runs analogously and corresponds to the remaining elongation 2(x At x a dicontinuity, denoted with 83, is present, i.e. the elongation suddenly rises from zero to the value 83, then remains constant up to x and then again in point 84 rises abruptly to the final value 85 (maximum value). Thus FIG. 12 also illustrates that by piecing together circular arcs to form a transition curve, as this has already been proposed in various cases, a stepped course for the elongation is obtained, i.e. an extremely unfavorable, abrupt stress of the strand at each transition from a straight direction to the first circular arc, or between the individual circular arcs, respectively, as well as at the end of the transition curve at the transition to the strand guiding in form of a circular arc.

FIG. 13 is an analogous illustration, if the transition curve follows an ellipse. The curve 86' for the elongation alteration rises very slowly at the beginning of the bending zone (x i.e. at first almost linearly, but then in the last section, it ascends very strongly according to the form of a parabola up to a maximum value 87 and from there it descends abruptly according to the curve section 86" down to zero. Thus at x again a discontinuity is present which is not desirable. The curve 88 for the elongation runs analogously. At x a discontinuity from zero to an infinite value 89 is present, from where the curve 88, similar to the curve 86, runs up to a maximum value 90, where a buckle exists. Here as well. the strand is exposed to a sudden load at x and at x it is abruptly freed from this load. If one uses an ellipse as transition curve, the conditions are better than if one uses circular arcs, but optimum bending conditions cannot be achieved.

FIG. 14 is an analogous illustration for the of a general parabola of a higher order as transition curve. The curve 91 for the elongation alteration starts from a zero value at x first ascends very strongly and afterwards, running gradually flatter, ascends to a maximum value 92, where a dicon tinuity is present, from which an abrupt descent to zero occurs (x The curve 93 for the elongation ascends from x to x gradually up to the maximum value, where a buckle exists.

Finally, FIG, ls illustrates the conditions when a clothoid is used as transition curve. The curve 94 for the elongation alteration has at x a discontinuity from zero to a finite value 95 and, at the end at 1: a maximum value 96, where a second discontinuity is present, where the curve descends to zero. The curve 97 for the elongation starts at x from zero and ascends very rapidly and fairly evenly up to a maximum value, where again a buckle ispresent. The clothoid as transition curve is thus comparable to an ellipse and also causes unfavorable conditions for bending a strand.

Hyperbola and catenary have at the transition from the straight line into the bending zone finite radii of curvature; there also dicontinuities occur.

The FIGS. 16 and 1 illustrate a straightening device according to the inventioh, wherein the strand 1 with the inner radius R, 7800 mm and the thickness d 200 mm enters the straightening device, the strand being elongated at the inner side and upset (compressed in longitudinal direction) at its outer side, thus being straightenedi The strand 1 leaves in the direction of the arrow with a radius R =R The straightening device consists of the rollers 100, 101, causing the straightening, and of the rollers 98, 99, accommodating the reaction and supporting forces. The straightening forces are marked with P 00, P101 in FIG. 17, and the bearing forces are denoted with A,, B 98 denotes the tangent point of the roller 98 at the inner side of the strand, which tangent point forms the zero point of a coordinate system, whose positive y-axis is marked with 102 and whose positive x-axis is denoted with 103 (the x-axis 103 is a tangent to the curve 111 (y (x,-)) in the point 98'). On the y-axis 102, which at the same time is the perpendicular through the point 98', lies the center of curvature M of the circular arc strand guiding with the inner radius R,,. 104, 105, 106 also denote perpendiculars in which lie the lines of influence of the bearing force B and of the straightening forces P P respectively. 107', 108' are intersection points of the perpendiculars 102, 104 with the outer transition curve 112, equidistant to the inner transition curve 111, and 109' and .110 are analogous intersection points on the curve 111. The distance a to the intersection point 109' on the x-axis 103 is given and measures 530 mm. The distance-b to the intersection point 110' from the intersection point 109' on the x-axis 103 is also given and measures I80 mm, and c, the distance between the tangent point 99 of the roller 99 and the intersection point 110' measures 570 mm. Thus the length of thestraightening zone X',, 1280 mm. Sought is Y ",'the course of the curve 1 11 according to the'functi'orty (x3 and the "course of the equidistant curve 112 as well as the intersection angles a a 01 of the perpendicillars 105, 106, 104 to the y-axis 102, which are important for the construction of the straightening device".

' FIGS. 18 and 19 are illustrations analogous to FIGS. 7 and 8. For the calculation of the curves for'the elongation alte'rat ion, andelongation, the equations 2, 5, 6, 7, 8, 9', 19 and20 and the modified further equations maybe used. L

The equations 13a. 14a refer to the inner side of the strand, and the equations 15a, 16a, l7a, 18a, refer to the outer side of the strand. The calculation is substantially the same as the calculation of a bending zone. The characteristic of the straightening device may be observed from the above mentioned FIGS. 18 and 19. 1 l3 denotes the course in form of a trapezoid of a standardized moment line or of a curve which is similar to the alteration of the curvature of the elongation alteration 6' (x respectively. The curve 113 defined by the points 98' (1: 114, 115, 116 (x;,) encloses together with the x-axis 103 the area 117. This area 117 is FLE. The upset curve, drawn true to scale, for the elongation alteration 2 (x,) is denoted with 113' in FIG. 19. It has two real zero points in the points 98', 116 and two maximum values 114', 115'. The pertaining curve 118 for the elongation 20g) passes from a horizontal branch 118' via a first point of inflexion 119 into an inflexion tangent 118" and via a second point of inflexion 120 as well as a maximum value 121, which corresponds to the maximum elongation, into a second horizontal branch 118'. 103 denotes a tangent to the point 121, a tangent coinciding with this branch 118', parallel to the x-axis 103, which is, as has been mentioned before, a characteristic of the course of the curve 118 for the elongation.

As is illustrated in FIG. 16, additional pairs of rollers 122 for guiding and supporting the strand skin may be arranged in the straightening device, if in a continuous casting plant, this straightening device is used for straightening a cast strand with a still liquid core. The roller pairs 122 may then be arranged floating,which means that, while maintaining their relative distance d in direction of the perpendiculars through the centers of the rollers, they are mounted to be freely movable. Further, guiding rollers are denoted with 107, 108, 109, 110. They, like the roller pairs 122, do not take part in the proper straightening of the strand. The same applies analogously to the stationary roller pairs, denoted with 123 of the circular arc-shaped strand guiding and to the stationary roller pairs denoted with 124 of the horizontal strand guiding with the radius R' R' The results of the numerical calculation are contained in the tables 7, 8, 9, table 7 containing numerical values for the curve 113' (elongation alteration) and 118 (total elongation) as well as for the current radius R, of the curve 1 l 1. Table 8 contains the numerical values for Y J for the curve 111 and its ascent y, again expressed as tangent of a, so that the tangent points of the rollers at the inner side of the strand and the inclination angles a ,a ,a,,, result immediately therefrom. Table 9 contains the coordinates x,", y," of the rollers 98, 109, 110, 99 at the inner side of the strand, and the coordinates E 7," of the two straightening rollers 100 and 101 at the outer side of the strand.

Table 7:

traightening zone (FIG. 16 19) Distance on elongation total radius R J the x-axis alteration elongation mm mm 103 "(IIIIII 0 (=x,) 0 0 7 800 50 0.169 0.00417 7 825 100 0.338 0.01667 7 903 0.506 0.03751 8 035 200 0.675 0.06668 8 228 250 0.844 0.10416 8 490 300 1.012 0.14997 8 833 350 1.178 0.20406 9 277 400 1.345 0.26644 9 846 450 1.512 0.33710 10 583 500 1.682 0.41602 11 547 530 (=Xm) 1.78 0.46731 12 274 550 1.78 0.50250 12 828 600 1.78 0.59039 14 458 650 1.78 0.67821 16 560 700 1.78 0.76595 19 376 710 14) 1.78 0.78349 20 058 750 1.65 0.851 16 23 207 800 1.49 0.92875 28 305 Table 71-Continued traightening zone (FIG. 16 19) Distance on elongation total radius R j the x-axis alteration elongation mm mm 1051 Mum Table 8:

Straightening zone (FIG. 16 19) Distance on Y Y j the x-axis mm mm (tan a 0 (=x,) 0 0 50 0.1602 0.0064 100 0.6396 0.0128 150 1.4353 0.0190 200 2.5418 0.0252 250 3.9521 0.0312 300 5.6567 0.0369 350 7.644 0.0424 400 9.901 0.0477 450 12.411 0.0526 500 15.158 0.0572 530 (=X 16.911 0.0597 550 18.121 0.0613 600 21.278 0.0649 650 24.609 0.0682 700 28.091 0.0710 710 (=x 28.803 0.0715 750 31.701 0.0734 800 35.420 0.0753 850 39.227 0.0769 900 43.105 0.0782 950 47.039 0.0791 1000 51.014 0.0798 1050 55.020 0.0803 1 100 59.046 0.0807 1 150 63.084 0.0808 1200 67. 1 29 0.0809 1250 71.177 0.0810 1280 (=x=,)=x' 73.606=Y" 0.0810

Table 9:

Straightening zone (FIG. 16 19) position of the x," Y," roller axes mm mm a) on the inner side roller 98 0 50 roller 109 527.02 66.82 roller 1 706.43 78.67 roller 99 1275.96 123.44

mm mm b) on the outer side roller 100 544.89 23264 roller 101 727.83 220.56

FIG. 20 is a synoptical drawing for a bending device according to FIGS. 5 and 6, and for a straightening device according to FIGS. 16 and 17, which are provided in a continuous steel casting plant, a circular arcshaped strand guiding with an outer radius R 8000 mm and an inner radius R, 7800 mm being arranged between the bending device and the straightening device. The thickness of the strand d 200 mm, and the center of curvature is denoted with M. From FIG. 20, the position of the coordinate systems can be seen. The coordinate system x, y with the axes 59, 53 has its origin at the outer side at the beginning of the bending zone in point 47'. The coordinate system x, y with the axes 103, 102 has its origin at the inner side at the beginning of the straightening zone in the point 98'. In the bending zone, the inner side of the strand is upset (compressed in longitudinal direction), while in the straightening zone, the outer side of the strand is upset.

A further calculation magnitude for the construction of a continuous casting plant according to FIG. 20, in which the strand is gradually diverted, i.e. bent from the vertical into a circular arc and from there straightened into a horizontal, is 01 41,, may be calculated from the relationship 21 wherein the angles a and 01 are to be seen from FIGS. 5 and 17. Thus, these angles are the inclination angle of the perpendicular 58 at the end of the bending device in relation to the horizontal, the y-axis 53, and the inclination angle of the perpendicular 102 at the beginning of the straightening device in relation to the vertical, which angle is identical with the inclination angle of the perpendicular 104 in relation to the y-axis 102.

It is advantageous that in bending of continuously cast steel strands with a liquid core and a solidified strand skin the maximum value of the elongation alteration does not exceed 0.0025 %/mm. In the straightening process, when the strand skin is naturally thicker and a little cooler, the maximum value of the elongation alteration should not exceed 0.0030 %/mm.

Furthermore, it is suitable, if the longitudinal extension X of the bending zone or X of the straightening zone, respectively, each measured in the direction of the pertaining x-axis, measures 1/7 up to US of the radius of curvature R and R respectively, of the circular arc.

Moreover, it is advantageous that the inclination angle a 01 of the perpendicular 35, 58 at the end of the bending zone in the transition point 20', 48' toward the circular arc to the y-axis 26, 53 amounts to 3 to 10, preferably to 5 to 7. The same applies for the inclination to the y-axis 102 of the perpendicular 104 at the transition point 99' from the straightening zone toward the horizontal.

A particularly advantageous embodiment of the invention in a continuous casting plant consists in that all the bearing and bending forces, both within the bending device and within the straightening device are transmitted onto the strand advantageously by means of non-driven rollers, wherein, for the adjustment to different strand thicknesses d, at least the rollers at the inner side of the strand are displaced parallel in relation to the strand skin, in direction of the perpendiculars to the points of impact of the bending forces in the bending device and of the bearing forces in the straightening device.

The essential technical advantage of the invention consists in that the probability of the occurrence of fissures caused by the dynamic deformation of the strand is minimized, particularly in so-called rapid casting 

1. In a continuous casting plant for strands comprising a mold and means defining a secondary cooling zone following the mold, the secondary cooling zone including guiding rollers and bending rollers arranged to define a bending zone of a certain extent, the bending rollers being arranged along a transition curve from the vertical direction into a circular arc, the improvement comprising that the bending rollers are arranged along a curve corresponding to the differential equation
 2. A plant according to claim 1, further comprising straightening rollers arranged to define a straightening zone of a certain extent, the straightening rollers being arranged along a transition curve from said circular arc into the horizontal, wherein the improvement also comprises that the straightening rollers are arranged along a curve corresponding to the differential equation
 3. A plant according to claim 1, wherein the maximumly permissible elongation D equals 50d/RE, d standing for the thickness of the strand, and RE being as defined in claim
 1. 4. A plant according to claim 1, wherein the bending rollers are arranged at points determined by a discontinuity-free course of the function of the change in elongation phi '' (xj).
 5. A plant according to claim 4, wherein said discontinuity-free course is substantially trapezoidal.
 6. A plant according to claim 4, wherein said discontinuity-free course is substantially circular-arc shaped.
 7. A plant according to claim 4, wherein said discontinuity-free course is substantially parabola-shaped.
 8. A plant according to claim 1, wherein the bending rollers are arranged at points determined in the following way: a. depending on the number of bending rollers, a function phi '' (xj) of the change in elongation graphically represented by a discontinuity-free curve is determined, b. said function is integrated to the function phi (xj) within the limits O to XE, then c. multiplying the integrated value by 1/RE, a transforming factor
 9. A plant according to claim 1, wherein the vertical component of the length XE of the bending zone amounts to 1/7 to 1/5 of the radius of curvature, RE, of the circular arc.
 10. A plant according to claim 1, wherein a perpendicular to the transition curve at the transition point from the end of the bending zone to the circular arc is inclined from 3* to 10* relative to the y-axis of the coordinate system having its origin at the beginning of the transition curve and its x-axis tangent to the transition curve.
 11. A plant according to claim 10, wherein said inclination is from 5* to 7*.
 12. In a continuous casting plant for strands, comprising a mold and means defining a secondary cooling zone following the mold, the secondary cooling zone including guiding rollers and straightening rollers arranged to define a straightening zone of a certain extent, the straightening rollers being arranged along a transition curve from a circular arc into the horizontal, the improvement comprising that the straightening rollers are arranged along a curve corresponding to the differential equation
 13. A plant according to claim 12, wherein the maximumly permissible elongation D equals 50d/RE, d standing for the thickness of the strand and RE being as defined in claim
 12. 14. A plant according to claim 12, wherein the straightening rollers are arranged at points determined by a discontinuity-free course of the function of the change in elongation phi '' (xj).
 15. A plant according to claim 14, wherein said discontinuity-free course is substantially trapezoidal.
 16. A plant according to claim 14, wherein said discontinuity-free course is substantially circular-arc shaped.
 17. A plant according to claim 14, wherein said discontinuity-free course is substantially parabola-shaped.
 18. A plant according to claim 12, wherein the straightening rollers are arranged at points determined in the following way: a. depending on the number of straightening rollers, a function phi '' (xj) of the change in elongation graphically represented by a discontinuity-free curve is determined, b. said function is integrated to the function phi (xj) within the limits O to XE, then c. multiplying the integrated value by 1/RE, a transforming factor
 19. A plant according to claim 12, wherein the x-axis component X''E of the length of of the straightening zone amounts to 1/7 to 1/5 of the radius of curvature RA of the circular arc.
 20. A plant according to claim 12, wherein a perpendicular to the transition curve at the transition point from the straightening zone to the horizontal is inclined from 3* to 10* relative to the y-axis of the coordinate system having its origin at the beginning of the transition curve and its x-axis tangent to the transition curve.
 21. A plant according to claim 20, wherein said inclination is from 5* to 7*. 